Fauns and Satyrs, while originally quite different, have often been fused together, both in the original myths and modern fiction. (Which is why they share a page.) Both are human from the waist up, but fauns have the legs of deer, while satyrs have the legs of goats. Both may have pointed ears, or horns, or both — and for some reason, both are much likelier to be male.
Originally, Satyrs had the tail and ears of either horses or donkeys, though later they acquired goat legs, becoming almost identical with fauns. In earlier Greek art they were portrayed as quite ugly, though later they became more youthful and graceful. They were associated with sex, wine, and pipe playing. They also laughed at everything, and had reverence for none but their patron, Pan, which is why their name is the source of the word "Satire."
A faun, more akin to deer than to goats, would be the satyr's more gentle and retiring cousin, preferring the forest to the field. They live in harmony with nature, and are generally just low-key and peaceful.
The most famous of all goat people is the great god Pan, and his Roman counterpart Faunus/Inuus. Pan was the god of all wild creatures, and had a trickster streak a mile wide. Despite his wildness and temper (his shouts would inspire pan-ic in all who heard him) he was a true friend to shepherds and little critters. And if your Satyr lives underground in a cavern full of flame, well, he might be a bit nastier...
See also Our Centaurs Are Different.
Caesar Clown's Gas Mask Mooks in One Piece are actually this. They used to be fully human, but because they lost the use of their legs due to the poisonous gas on Punk Hazard, they were given new animal legs by Trafalgar Law.
Pan's Labyrinth has a faun, although in a twist authorized by the Rule of Creepy, he seems to be made mostly of rotting wood, thus representing a fusion of both the animal and vegetable realms of nature.
Mr. Tumnus, a faun in The Chronicles of Narnia. Also, other fauns are mentioned throughout the series, along with satyrs. The difference between them is that fauns have long tails, and satyrs have goat tails. The films expanded this difference, and made fauns human from the waist up with regular got tails, and satyrs looking more like human-sized goats that walk on their hind legs.
In Xanth, Fauns love chasing nymphs and simulating summoning the stork with them.
Matthew in Satyrday is ages old, but is quite irresponsible, with a mind of a child. Bonus points for having a pipe and being quite skilled with it.
Nathaniel Hawthorne wrote a novel called The Marble Faun, though the faun there is the statue by Praxiteles, which is almost totally human in form. The faun has a human counterpart, however, in the novel's Donatello (no, not that one), Count of Monti Beni.
In Wolfwinter by Thomas Burnett Swann, one of his many stories drawing on classical mythology, a human woman searches for the satyr who fathered her child. Swann's portrayals of non-humans can involve Blue and Orange Morality.
Satyrs and Fauns are considered as the same things.
Hedge mentioned that "Fauns are Roman." Seeing as how Roman demigods are quite different from their Greek counterparts, it wouldn't be surprising if this was the case with other magical creatures. Possibly justified. They could have indeed become more or less the same if people began to believe that they were with different names. In the first series, it was said that the Titans Helios and Selene faded after the Romans reassigned their duties over the sun and moon to Apollo and Artemis, so it's not unprecedented for them to have been blended them together into the same creature/species.
And as of Son Of Neptune, we've seen fauns, and they are, in fact, not the same as satyrs, to Percy's confusion.
Satyrs appear in the Fablehaven books. They have some of the qualities of the original satyrs (they're often seen trying to impress girls, and at one point, they offer Seth wine), but they're also TV addicts.
In the book Through the Ice the character Rame is in self-exile from the community of satyrs because he considers himself a faun. While he is less obsessed with sex and thus somewhat more acceptable to humans, his fellow satyrs treat him like a pervert.
The Orphan's Tales series features the Gaselli, who are obviously enough more related to Gazelle, but have a similar motif. A race of nature oriented beings with cloven hooves. They tend to be more interested in food than sex however. Eshkol and her family represent more traditional version of this trope.
Kate Forsyth's Witches of Eileanan universe has "satyricorns." While they mostly look like the traditional model in appearance (except for being seven feet tall, having a variable number and arrangement of horns, and having six mammaries) and behavior (albeit towards the nastierside), they zigzag the trope by being primarily female.
Edmund Spenser's The Faerie Queene features satyrs, first as a group of surprisingly pious forest dwellers who even have a half-satyr knight living among them, Sir Satyrane. Later, a separate band of satyrs are depicted as the usual riotous sex maniacs.
The Everworld gang meet a bunch of satyrs at one point. When they get a little too rowdy with April, David cuts one in half, which not only fails to kill him, but also lets the gang discover that they don't really have organs. The satyrs themselves are too drunk to care, the bisected one complaining that the wine is just pouring onto the ground. The bottom half follows them around, and is traded to a leprechaun for entry into their city.
Satan and his demons are often depicted similarly to fauns, (though never in The Bible) having cloven hooves and horns, with the addition of a long tail with a arrow head shape at the end, (sometimes) bat wings, and a bright red color. See Big Red Devil.
Dungeons & Dragons has satyrs that are a combination of the fauns and satyrs of Greek Mythology. The 3.5 edition of Deities and Demigods also offers stats on fauns as a separate, less powerful, race. Also worth mentioning are the Ibixians, a race of humanoid goats.
In the Theros Block, satyrs are first introduced as a valid creature type. They are depicted basically as duplicitous, almost downright evil hedonists: they depict themselves as happy, joyful free spirits that provide entertainment for free, but their more secret revels are violent and extremely depraved (there's a reason said revels are called called "Bakkeia"), and they lure humans into being their servants, humiliating them all the while. Basically, a very, very dark version of The Fair Folk, which is no small accomplishment, given the competition.
Xenagos is a satyr planeswalker from Theros that used to be like those above, but after ascending and exploring the multiverse he realised he was utterly insignificant. So he returns to Theros with one thing in mind...
Shadowrun. Both male and female satyrs have large curling horns like a bighorn sheep. They're implied to be the Awakened version of the wild goat. Their saliva can ferment sugary liquids into alcohol. Another type of Satyr exists, as a racial variant of the Ork Metatype.
Beastmen in Warhammer are about all the darkest parts of this put together.
Broos in RuneQuest. There are also actual satyrs. Mistaking the two for one another is not a good idea. The Broos are chaos creatures that breed by raping any creatures that are available (all Broos are male), and their offspring takes features of those creatures. Farm animals, especially goats are apparently most easily available for them.
In GURPS Dungeon Fantasy, Saturs are a possible character race, which includes females as well.
Both Satyrs and Fauns are part of the Fae in Pathfinder. Fauns are actually born from the copulation of the One-Gender Race Satyrs with particularly Good women. Unlike their fathers, Fauns can in fact be female (though female Fauns are rare).
The famous Greek tragedies originally came in tetralogies. The first three were traditional tragedies following some sort of story arch. The final play was a Satyr play which satirized the first three (though "satyr" and "satire" aren't actually etymologically related) and also made tasteless jokes about well-known people. They were so-called because the actors wore fur leggings and big fake leather cocks, like satyrs. Of the hundreds of ancient satyr plays only one still exists today, Euripides' The Cyclops, which is indeed a ribald satirical depiction of the cyclops island passage of The Odyssey.
Satyrs also show up in Dragon Quest VIII, in the bestiary. They use their flutes as a weapon.
Satyrs in the Warcraft universe are half-demonic corrupted night elves, and Fauns are half-daughters of the Demigod Cenarius.
Spyro 2: Ripto's Rage!/Gateway to Glimmer has both fauns and satyrs. In addition to primary character Elora, there are humanoid-looking fauns and satyrs in the Fracture Hills, and smaller more monster-y looking fauns in the Magma Cone. Exactly what differentiates them is not entirely clear. It's also worth noting that the Fracture Hills fauns, and to a lesser extant Elora, aren't all that humanoid to begin with. The ones in Fracture Hills in particular look more like anthropomorphic wolves with goat legs. Also in Fracture Hills the fauns are all female and the satyrs are male, while in Magma Core the fauns seem to be mostly male.
Diablo has Goatmen, which are actually demons and not related to either goats or humans.